Breaking the Stick
Breaking the Stick
Breaking the Stick
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G13 BREAKING A STICK #1<br />
G 1 3<br />
Capsule Lesson Summary<br />
Given two line segments, construct as many essentially different triangles as possible<br />
with each side <strong>the</strong> same length as one of <strong>the</strong> line segments. Discover a rule relating <strong>the</strong><br />
length of <strong>the</strong> segments to <strong>the</strong> number of different triangles. Repeat this activity with<br />
three given line segments and discover <strong>the</strong> Triangle Inequality.<br />
Materials<br />
Teacher<br />
• Chalkboard compass<br />
• Straightedge<br />
Student<br />
• Unlined paper<br />
• Straightedge<br />
• Compass<br />
• Metric ruler<br />
• Worksheets G13(a) and (b)<br />
Exercise 1<br />
Draw two line segments on <strong>the</strong> board, one about 25 cm long and <strong>the</strong> o<strong>the</strong>r about 40 cm long.<br />
T: These two line segments have different lengths. Let’s draw a triangle using a compass and<br />
a straightedge. Each side of <strong>the</strong> triangle must be <strong>the</strong> same length as one of <strong>the</strong>se line<br />
segments.<br />
Invite students to perform <strong>the</strong> construction at <strong>the</strong> board while <strong>the</strong> class discusses <strong>the</strong> technique.<br />
(See IG-V Lesson G12 Polygons #2 for a description of compass and straightedge constructions.)<br />
Be sure students realize that although a meter stick or metric ruler may be used for drawing straight<br />
line segments in <strong>the</strong>se constructions, it should not be used for measuring. There are four essentially<br />
different triangles that can be constructed depending on <strong>the</strong> number of each length side chosen:<br />
three short; two short and one long; one short and two long; and three long.<br />
If an equilateral triangle is constructed ei<strong>the</strong>r with three short or with three long sides, comment on<br />
its correctness, and <strong>the</strong>n introduce <strong>the</strong> restriction that each of <strong>the</strong> two lengths must be used at least<br />
once in each triangle. During <strong>the</strong> discussion, introduce <strong>the</strong> idea that <strong>the</strong> orientation of <strong>the</strong> triangle on<br />
<strong>the</strong> board does not alter it in any way important to its identity with respect to <strong>the</strong> choice of sides.<br />
Thus, <strong>the</strong>se four triangles should all be thought of as essentially <strong>the</strong> same.<br />
IG-VI<br />
G-57
G 1 3<br />
Once <strong>the</strong> class is secure in <strong>the</strong> triangle construction technique and <strong>the</strong>re are two triangles (short-short-long<br />
and long-long-short) on <strong>the</strong> board, distribute copies of Worksheet G13(a) which shows five pairs of<br />
line segments. Direct students to construct as many triangles as possible with each pair of segments.<br />
Each side of a triangle must be <strong>the</strong> same length as one member of <strong>the</strong> pair, and each line segment of<br />
<strong>the</strong> related pair must be used at least once. The pairs are denoted A, B, C, D, and E. Instruct students<br />
to mark each triangle with <strong>the</strong> same letter as <strong>the</strong> pair of segments used to construct it. As students fill<br />
up <strong>the</strong> space on <strong>the</strong> worksheet, provide <strong>the</strong>m with unlined paper.<br />
Encourage accurate and careful constructions. Allow time for experimentation and for <strong>the</strong> conviction<br />
to grow that in no case are more than two triangles possible. Encourage students to try to formulate<br />
a rule to predict <strong>the</strong> possibility of one or two triangles, and <strong>the</strong>n to draw pairs of segments to test <strong>the</strong><br />
rule. Now students may use <strong>the</strong> metric rulers for measuring in order to fur<strong>the</strong>r test a rule based on<br />
relative lengths of <strong>the</strong> line segments.<br />
Ask several students to measure, to <strong>the</strong> nearest centimeter, <strong>the</strong> lengths of <strong>the</strong> line segments on <strong>the</strong><br />
worksheet. Collect <strong>the</strong> results in a table on <strong>the</strong> board.<br />
Lead <strong>the</strong> discussion to elicit a rule for deciding when two triangles are possible. There are at least<br />
two good ways to state this rule:<br />
• The short segment must be more than half as long as <strong>the</strong> long segment.<br />
OR<br />
• Twice <strong>the</strong> length of <strong>the</strong> short segment must be more than <strong>the</strong> length of <strong>the</strong> long segment.<br />
Check students’ understanding of <strong>the</strong> rule by<br />
listing several pairs of lengths and asking for<br />
<strong>the</strong> number of possible triangles.<br />
Exercise 2<br />
Pose a triangle construction problem where three different length line segments are given.<br />
T: Now let’s use three segments of different lengths to draw triangles. Each segment must be<br />
used once in a triangle.<br />
G-58 IG-VI
G 1 3<br />
Distribute copies of Worksheet G13(b). Proceed as with Worksheet G13(a). After <strong>the</strong> individual<br />
work, collect <strong>the</strong> results on <strong>the</strong> board.<br />
Discuss <strong>the</strong> results with <strong>the</strong> purpose of formulating a rule for <strong>the</strong> possibility of constructing a<br />
triangle. This is a good formulation:<br />
• The sum of <strong>the</strong> lengths of <strong>the</strong> two shorter segments must be more than <strong>the</strong> length of <strong>the</strong><br />
longest segment. (Triangle Inequality)<br />
Ask students to apply <strong>the</strong> rule to several sets of lengths, deciding whe<strong>the</strong>r or not a triangle can be<br />
formed.<br />
IG-VI<br />
G-59
G 1 3<br />
G-60 IG-VI
G14 BREAKING A STICK #2<br />
G 1 4<br />
Capsule Lesson Summary<br />
Pose <strong>the</strong> problem of finding <strong>the</strong> probability that if we break a stick in two places,<br />
choosing those places randomly, <strong>the</strong> three resulting pieces will form a triangle. Examine<br />
many cases to decide conditions for success and for failure. Rephrase <strong>the</strong> Triangle Inequality.<br />
Materials<br />
Teacher<br />
• 20 cm stick or straw<br />
• Straightedge<br />
• Blacklines G14(a) and (b)<br />
Student<br />
• Colored pencils<br />
• Compass<br />
• Metric ruler<br />
• Unlined paper<br />
• Sheet with 20-cm line segments<br />
• <strong>Breaking</strong> points table<br />
• Worksheets G14* and **<br />
Advance Preparation: Locate a stick that breaks easily (e.g., balsa wood) or a straw that you can cut to<br />
20 cm length. For students, use Blackline G14(a) to make copies of a sheet with 20 cm line segments; use<br />
Blackline G14(b) to makes copies of <strong>the</strong> table for recording broken points.<br />
Display a stick (or straw) 20 cm long.<br />
T: Suppose we have a stick 20 cm long and we break it in two places. What is <strong>the</strong> probability<br />
of being able to make a triangle with <strong>the</strong> three pieces if <strong>the</strong> breaking points are chosen at<br />
random?<br />
Let students freely discuss <strong>the</strong> problem. Some students may wish to estimate <strong>the</strong> probability;<br />
o<strong>the</strong>rs to suggest ways of finding <strong>the</strong> probability; and o<strong>the</strong>rs to discuss how to break <strong>the</strong> stick<br />
at two randomly chosen breaking points. At some time during <strong>the</strong> discussion, let <strong>the</strong> class select<br />
two breaking points on <strong>the</strong> stick. Make <strong>the</strong> breaks and try to make a triangle with <strong>the</strong> three pieces.<br />
Distribute copies of Blackline G14(a) to pairs of students.<br />
T: There are five 20-centimeter line segments drawn on this sheet. Each has two breaking<br />
points indicated by dots. In each case, decide whe<strong>the</strong>r a triangle can be formed with <strong>the</strong><br />
three pieces. Use a compass and a ruler for a straightedge.<br />
Note: If you prefer, give student pairs five 20-cm pieces of string to cut at <strong>the</strong> breaking points indicated<br />
on <strong>the</strong> sheet. Then <strong>the</strong>y can attempt to make triangles with <strong>the</strong> pieces of string straightened as line<br />
segments.<br />
Help students who have difficulty getting started. When most student pairs have completed four or<br />
five of <strong>the</strong> problems, discuss <strong>the</strong>m collectively. The following illustrations show two similar methods<br />
of construction.<br />
IG-VI<br />
G-61
G 1 4<br />
T: You were able to make triangles with <strong>the</strong> segments in A and in C but not with those in B, in<br />
D, and in E. Why?<br />
S: The two short pieces toge<strong>the</strong>r must be longer than <strong>the</strong> longest piece.<br />
T: Let’s measure to check what you are saying.<br />
S: The three lengths in A are 5 cm, 8 cm, and 7 cm. 5 + 7 is more than 8, so we can make a<br />
triangle.<br />
S: The three lengths in B are 3 cm, 5 cm, and 12 cm. 3 + 5 is less than 12, so we cannot make<br />
a triangle.<br />
T: In C, what are <strong>the</strong> three lengths?<br />
S: 9 cm, 2 cm, and 9 cm.<br />
T: So we have to compare 9 + 2 to 9 since we must use a 9 cm segment as a longest piece.<br />
S: 9 + 2 is more than 9, so we can make a triangle. It will be a narrow triangle.<br />
In a similar manner, discuss D and E.<br />
Begin a table on <strong>the</strong> board similar to that on Blackline G14(b). Provide students with a copy. Draw a<br />
20-cm line segment on <strong>the</strong> board, marking one end 0 and <strong>the</strong> o<strong>the</strong>r 20. Then place breaking points at<br />
5 cm and 14 cm.<br />
T: If we break <strong>the</strong> 20-cm stick<br />
at 5 cm and at 14 cm, what<br />
would <strong>the</strong> lengths of <strong>the</strong> three<br />
pieces be?<br />
S: 5 cm, 9 cm, and 6 cm.<br />
T: Could we make a triangle?<br />
S: Yes, 5 + 6 > 9.<br />
Record <strong>the</strong> information in <strong>the</strong> table. Instruct students to use <strong>the</strong> five 20-cm line segments on<br />
Blackline G14(a) to enter data in <strong>the</strong> table. Continue <strong>the</strong> table with o<strong>the</strong>r choices for <strong>the</strong> breaking<br />
points, occasionally letting students choose <strong>the</strong>m. Be sure to include an example in which one of <strong>the</strong><br />
three lengths is 10 cm and, hence, <strong>the</strong> sum of <strong>the</strong> lengths of <strong>the</strong> o<strong>the</strong>r two segments is exactly 10 cm,<br />
giving a failure.<br />
G-62 IG-VI
Your table should resemble this one.<br />
G 1 4<br />
T: If one of <strong>the</strong> breaks is at 4 cm, where could <strong>the</strong> o<strong>the</strong>r break be so that we could form a<br />
triangle? Why?<br />
S: At 11 cm. The lengths would be 4 cm, 7 cm, and 9 cm. 4 + 7 > 9.<br />
S: At 12 cm. The lengths would be 4 cm, 8 cm, and 8 cm. 4 + 8 > 8.<br />
Any breaking point between 10 cm and 14 cm would be a correct answer. Record this information<br />
in <strong>the</strong> table.<br />
Consider o<strong>the</strong>r possibilities such as having one breaking point at 13 cm, one at 10.5 cm, one at<br />
10.25 cm, and one at 10 cm. Only in <strong>the</strong> last case is it impossible to construct a triangle. Record<br />
<strong>the</strong> information in separate lines in <strong>the</strong> table, as illustrated here.<br />
T: When can we make a triangle? How long can <strong>the</strong> pieces be for success?<br />
Students may state <strong>the</strong> Triangle Inequality (as given in Lesson G13), but lead <strong>the</strong>m to also notice <strong>the</strong><br />
equivalent rule that each piece must be shorter than half <strong>the</strong> length of <strong>the</strong> stick (in this case, 10 cm).<br />
To elicit this rule, direct attention to <strong>the</strong> list of lengths in <strong>the</strong> table. Successes occur when exactly all<br />
lengths are less than 10 cm. In <strong>the</strong> discussion, someone might comment that if <strong>the</strong> longest length is<br />
more than 10 cm, <strong>the</strong>n <strong>the</strong> o<strong>the</strong>r two pieces toge<strong>the</strong>r must be shorter than 10 cm since <strong>the</strong> stick is<br />
20 cm long. So if one piece is longer than 10 cm, <strong>the</strong>re is a failure.<br />
IG-VI<br />
G-63
G 1 4<br />
Worksheets G14* and ** are available for individual work to provide practice determining<br />
successful choices of points. On <strong>the</strong> ** worksheet, explain that <strong>the</strong> open dots in <strong>the</strong> example are<br />
used only to indicate that certain points cannot be breaking points, whereas every point between<br />
<strong>the</strong> open dots could be a second breaking point.<br />
G-64 IG-VI
G15 BREAKING A STICK #3<br />
G 1 5<br />
Capsule Lesson Summary<br />
Establish <strong>the</strong> one-to-one correspondence between a point in a triangle and two breaking<br />
points on a stick. Use this correspondence to provide a means for randomly choosing two<br />
breaking points.<br />
Materials<br />
Teacher<br />
• Colored chalk<br />
• Meter stick<br />
• Grid board<br />
Student<br />
• Worksheets G15* and**<br />
Advance Preparation: Use Blackline G15 to make a grid, or prepare your grid board as indicated on this<br />
Blackline.<br />
On <strong>the</strong> board, draw a line segment about 1 m long, and refer to this as a “stick” in <strong>the</strong> following<br />
discussion.<br />
T: What is <strong>the</strong> “breaking a stick” problem?<br />
S: If you break a stick in two places to make three pieces, sometimes <strong>the</strong> three pieces can be<br />
used to form a triangle, and sometimes <strong>the</strong>y cannot. We want to know <strong>the</strong> probability of<br />
forming a triangle if <strong>the</strong> breaking points are chosen at random.<br />
T: When will <strong>the</strong> three pieces form a triangle?<br />
S: When <strong>the</strong> shorter two pieces toge<strong>the</strong>r are longer than <strong>the</strong> longest piece.<br />
T: With our 20-cm stick, could one of <strong>the</strong> pieces be 10 cm long?<br />
S: No, because toge<strong>the</strong>r <strong>the</strong> o<strong>the</strong>r two pieces would be 10 cm long. When we try to make a<br />
triangle, <strong>the</strong> two shorter pieces collapse to a line segment 10 cm long.<br />
Students often use <strong>the</strong>ir hands in describing this situation.<br />
T: Could one of <strong>the</strong> pieces be longer than 10 cm?<br />
S: No, <strong>the</strong>re would be even less of <strong>the</strong> stick left<br />
for <strong>the</strong> o<strong>the</strong>r two pieces.<br />
T: So what can we say about <strong>the</strong> length of each of three pieces that will form a triangle?<br />
S: Each is shorter than 10 cm.<br />
Write this requirement on <strong>the</strong> board for emphasis. Mark <strong>the</strong> midpoint of <strong>the</strong> line segment on <strong>the</strong><br />
board and label it 10 cm.<br />
T: The problem involves a question about probability. If we choose <strong>the</strong> breaking points at<br />
random, what is <strong>the</strong> probability that <strong>the</strong> three pieces will make a triangle? How can we<br />
choose two breaking points at random?<br />
IG-VI G-65
G 1 5<br />
Let students suggest and discuss devices (spinners, darts, and so on) that might be used for choosing<br />
<strong>the</strong> breaking points.<br />
T: Name some possible breaking points and let’s see if <strong>the</strong>y are successes or failures.<br />
S: 5 cm and 13 cm.<br />
Locate <strong>the</strong> breaking points. Mark <strong>the</strong>m with s and label <strong>the</strong>m.<br />
T: If we break <strong>the</strong> stick at 5 cm and at 13 cm, would <strong>the</strong> resulting pieces form a triangle?<br />
S: Yes; <strong>the</strong> three pieces would be 5 cm, 8 cm, and 7 cm long—each is less than 10 cm long.<br />
Continue <strong>the</strong> activity until five or six pairs of breaking points have been suggested and discussed. In<br />
<strong>the</strong> next illustration, success (you can form a triangle) is shown in blue and failure (you cannot form<br />
a triangle) is shown in red.<br />
Refer to <strong>the</strong> first pair of breaking points (5 cm and 13 cm) listed on <strong>the</strong> board.<br />
T: Let’s record this pair of breaking points with a blue dot at a point on <strong>the</strong> grid. Where<br />
should we put <strong>the</strong> dot?<br />
There are two points that would be natural to<br />
use: (5, 13) and (13, 5). Mark each with a blue<br />
dot and connect <strong>the</strong> two dots with a segment.<br />
Continue by graphing <strong>the</strong> o<strong>the</strong>r examples of<br />
breaking points listed on <strong>the</strong> board. Use blue<br />
or red dots according to whe<strong>the</strong>r <strong>the</strong> points<br />
give a success or a failure.<br />
G-66<br />
IG-VI
Choose a student to come to <strong>the</strong> board. Every time<br />
you touch a point in <strong>the</strong> picture, ask <strong>the</strong> student to<br />
touch <strong>the</strong> o<strong>the</strong>r point that could represent <strong>the</strong> same<br />
breaking points. Repeat <strong>the</strong> activity several times.<br />
G 1 5<br />
T: What do you notice about <strong>the</strong>se pairs of points?<br />
S: The picture is symmetrical.<br />
T: Where would we place a mirror to see <strong>the</strong> symmetry?<br />
Invite a student to show where one would place a mirror.<br />
The student should trace <strong>the</strong> diagonal line segment that<br />
passes through <strong>the</strong> points (0, 0) and (20, 20).<br />
T: So that we have only one point for each pair of<br />
breaking points, let <strong>the</strong> first coordinate be for <strong>the</strong><br />
rightmost breaking point (label <strong>the</strong> horizontal axis)<br />
and let <strong>the</strong> second coordinate be for <strong>the</strong> leftmost<br />
breaking point (label <strong>the</strong> vertical axis). Where <strong>the</strong>n<br />
would dots for pairs of breaking points for <strong>the</strong> stick be?<br />
S: Below <strong>the</strong> diagonal line.<br />
T: But what about points on <strong>the</strong> diagonal line?<br />
S: They would be for <strong>the</strong> two breaking points being <strong>the</strong> same.<br />
T: That could happen if <strong>the</strong> breaking points were chosen at random. When <strong>the</strong> breaking<br />
points are <strong>the</strong> same, can we form a triangle with <strong>the</strong> pieces?<br />
S: No, <strong>the</strong>re are only two pieces.<br />
Redraw <strong>the</strong> diagonal line in red. Erase <strong>the</strong> connecting cords and <strong>the</strong> dots above <strong>the</strong> diagonal.<br />
Trace <strong>the</strong> line segment from (20, 0) to (20, 20).<br />
T: What can we say about <strong>the</strong>se points?<br />
S: They are for <strong>the</strong> rightmost breaking point being 20 cm. There would really be no break,<br />
because 20 cm is at <strong>the</strong> end of <strong>the</strong> stick. Also, <strong>the</strong>re would not be three pieces to form a<br />
triangle.<br />
Draw <strong>the</strong> line segment from (20, 0) to (20, 20) in red.<br />
Likewise, conclude that all points along <strong>the</strong> line segment<br />
from (0, 0) to (20, 0) should be red.<br />
Point to a grid point below <strong>the</strong> diagonal.<br />
T: How can we find which breaking points this<br />
point on <strong>the</strong> graph represents?<br />
IG-VI G-67
G 1 5<br />
Invite a student to demonstrate <strong>the</strong> technique at <strong>the</strong> board.<br />
The student should project <strong>the</strong> point onto each axis.<br />
For example:<br />
T: The stick is broken at 12 cm and at 15 cm. Let<br />
me show you a way in which we can represent<br />
<strong>the</strong> stick in <strong>the</strong> picture and show <strong>the</strong> breaking<br />
points on <strong>the</strong> stick. (Erase <strong>the</strong> labels of <strong>the</strong> axes.)<br />
Let this be <strong>the</strong> stick.<br />
Project down as we did before to get one breaking<br />
point. Project to <strong>the</strong> left, but stop at <strong>the</strong> diagonal<br />
line and project straight down to get <strong>the</strong> second<br />
breaking point.<br />
Be sure <strong>the</strong> students observe that this method of projection<br />
gives <strong>the</strong> same set of breaking points.<br />
T: Does this dot represent a success or a failure for<br />
making a triangle?<br />
S: Failure (<strong>the</strong> answer depends on <strong>the</strong> example).<br />
Color <strong>the</strong> dot red for failure or blue for success.<br />
Demonstrate <strong>the</strong> technique with several o<strong>the</strong>r points, tracing without drawing.<br />
T: Now let’s take two breaking points on <strong>the</strong> stick<br />
and find <strong>the</strong> point in <strong>the</strong> triangle that represents<br />
<strong>the</strong>m.<br />
Mark two breaking points on <strong>the</strong> stick, and ask for a<br />
volunteer to show how to find <strong>the</strong> corresponding<br />
point in <strong>the</strong> red triangle, or do so yourself.<br />
Repeat <strong>the</strong> activity with ano<strong>the</strong>r pair of breaking points.<br />
T: For every pair of breaking points we can find a point in <strong>the</strong> triangle, and for every point in<br />
<strong>the</strong> triangle we can find a pair of breaking points. This gives us a way to choose two<br />
breaking points randomly—we just need to choose one point in <strong>the</strong> triangle randomly.<br />
How could we choose a point in <strong>the</strong> triangle randomly?<br />
Let students make suggestions.<br />
G-68<br />
IG-VI
Shade <strong>the</strong> interior of <strong>the</strong> (red) triangle as you say,<br />
G 1 5<br />
T: We could smear honey everywhere inside <strong>the</strong><br />
triangle and set a fly loose in <strong>the</strong> room. The fly<br />
will choose a point at random on which to land.<br />
Worksheets G15* and ** are available for individual practice making projections.<br />
IG-VI G-69
G 1 5<br />
G-70<br />
IG-VI
G16 BREAKING A STICK #4<br />
G 1 6<br />
Capsule Lesson Summary<br />
Recall <strong>the</strong> problem of breaking a stick. Using <strong>the</strong> fact that no piece can be as long or<br />
longer than half of <strong>the</strong> stick and <strong>the</strong> idea of <strong>the</strong> “honey triangle” from Lesson G15, find<br />
<strong>the</strong> probability of getting three pieces that form a triangle when two breaking points are<br />
selected at random.<br />
Materials<br />
Teacher<br />
• Colored chalk<br />
• Meter stick<br />
• Grid board<br />
Student<br />
• Colored pencils<br />
• Worksheet G16<br />
Advance Preparation: Use Blackline G15 to make a grid, or prepare your grid board as indicated on this<br />
Blackline.<br />
Draw <strong>the</strong> “honey triangle” from <strong>the</strong> end of Lesson G15 on <strong>the</strong> grid.<br />
T: Who can recall <strong>the</strong> “breaking <strong>the</strong> stick” problem?<br />
S: If we break a stick at two points chosen at random, what is <strong>the</strong> probability that we can<br />
make a triangle with <strong>the</strong> resulting three pieces?<br />
T: How can we choose <strong>the</strong> two points at random?<br />
S: By letting a fly land on <strong>the</strong> honey triangle.<br />
Refer students to <strong>the</strong>ir copies of Worksheet G16 and as you give <strong>the</strong>se directions.<br />
T: Last week we found a one-to-one correspondence between pairs of breaking points on a<br />
stick and points in <strong>the</strong> honey triangle. Let’s mark some points in <strong>the</strong> triangle with blue or<br />
red dots. If a point corresponds to breaks resulting in three pieces that will form a triangle,<br />
draw a blue dot. O<strong>the</strong>rwise, draw a red dot.<br />
Allow about ten minutes for <strong>the</strong> students to<br />
mark red and blue points in <strong>the</strong> triangle on<br />
<strong>the</strong>ir worksheets. Invite students to mark red<br />
and blue points in <strong>the</strong> graph on <strong>the</strong> board.<br />
General areas of red dots and blue dots will<br />
become obvious. Encourage students to<br />
comment. If a point looks to be incorrectly<br />
colored, question <strong>the</strong> student who drew it and<br />
make necessary changes in color. After a while<br />
your picture will look similar to this one.<br />
T: Let’s look closer at <strong>the</strong> situation. If one<br />
of <strong>the</strong> breaking points is at 10 cm, can<br />
we ever make a triangle?<br />
IG-VI G-71
G 1 6<br />
S: No, because one of <strong>the</strong> pieces would be 10 cm long.<br />
Illustrate <strong>the</strong> two possible situations on <strong>the</strong> board.<br />
T: Which points in <strong>the</strong> triangle correspond to having one breaking point at 10 cm?<br />
Ask a student to show <strong>the</strong>m in <strong>the</strong> picture on <strong>the</strong> board.<br />
The points lie on two line segments. Draw <strong>the</strong>m in red.<br />
Point to <strong>the</strong> small triangle at <strong>the</strong> lower left.<br />
T: There are red dots in this region. Could <strong>the</strong>re be<br />
a blue dot in this region?<br />
S: No; <strong>the</strong> rightmost break would be at a number<br />
less than 10, making <strong>the</strong> piece on <strong>the</strong> right longer<br />
than 10 cm.<br />
S: If we are to get a triangle, both breaks cannot be<br />
on <strong>the</strong> same half of <strong>the</strong> stick.<br />
Illustrate <strong>the</strong> situation.<br />
If some students are having difficulty, choose several points in <strong>the</strong> lower left triangular region and<br />
illustrate where <strong>the</strong> breaks would be in each case. When <strong>the</strong> class is convinced that no blue dot<br />
belongs in that small triangle, color its interior red.<br />
Refer to <strong>the</strong> small triangle at <strong>the</strong> upper right.<br />
T: There are red dots in this region. Could <strong>the</strong>re be a blue dot in this region?<br />
S: No; <strong>the</strong> leftmost break would be at a number more than 10, making <strong>the</strong> piece on <strong>the</strong> left<br />
longer than 10 cm.<br />
S: Again, if we are to get a triangle, both breaks cannot be on <strong>the</strong> same half of <strong>the</strong> stick.<br />
G-72<br />
IG-VI
When <strong>the</strong> class is convinced that no blue dot belongs<br />
in <strong>the</strong> small upper-right triangle, color its interior red.<br />
G 1 6<br />
Point to <strong>the</strong> square inside <strong>the</strong> large triangle.<br />
T: In this region, we have some blue dots and<br />
some red dots. Is <strong>the</strong>re any pattern?<br />
Perhaps a student will indicate that blue dots seem to fall<br />
above <strong>the</strong> diagonal from (10, 0) to (20, 0) and red dots fall<br />
below it.<br />
T: Let’s look at <strong>the</strong> situation more carefully. If <strong>the</strong> rightmost<br />
break is at 15, where could <strong>the</strong> leftmost break be to yield a success?<br />
Illustrate <strong>the</strong> cases as students discuss <strong>the</strong>m.<br />
S: The leftmost breaking point cannot be at 5<br />
because <strong>the</strong>n <strong>the</strong> middle segment would be<br />
10 cm long.<br />
S: The leftmost breaking point cannot be at a<br />
number less than 5 because <strong>the</strong>n <strong>the</strong> middle<br />
segment would be longer than 10 cm.<br />
S: The leftmost breaking point can be at any<br />
number more than 5 but less than 10. For<br />
example, if <strong>the</strong> leftmost break were at 8,<br />
<strong>the</strong>n <strong>the</strong> three pieces would be 8 cm, 7 cm,<br />
and 5 cm long.<br />
T: Why does <strong>the</strong> leftmost breaking point have to be at a number less than 10? Why not at 10?<br />
Why not at 12?<br />
S: The leftmost breaking point cannot be at 10 because <strong>the</strong>n <strong>the</strong> left piece would be 10 cm<br />
long.<br />
S: The leftmost break cannot be at 12 because<br />
<strong>the</strong>n <strong>the</strong> left piece would be longer than 10 cm.<br />
For a situation in which <strong>the</strong> rightmost break is at 15,<br />
indicate success and failure points with a line segment<br />
partially red and partially blue. Mark <strong>the</strong> point (15, 5)<br />
with a red dot.<br />
IG-VI G-73
G 1 6<br />
Continue <strong>the</strong> activity, considering each whole number between 10 and 20 as a possible rightmost<br />
break. Your picture should look like <strong>the</strong> one below.<br />
Try o<strong>the</strong>r choices for <strong>the</strong> rightmost breaking point, such as 12.5 and 16.25.<br />
A success when <strong>the</strong> rightmost break is at 12.5 results if <strong>the</strong> choice for <strong>the</strong> leftmost break is a number<br />
more than 2.5 but less than 10.<br />
The leftmost break can be anywhere between 2.5 and 10.<br />
A success when <strong>the</strong> righmost break is at 16.25 results if <strong>the</strong> choice for <strong>the</strong> leftmost break is a number<br />
more than 6.25 but less than 10.<br />
The leftmost break can be anywhere between 6.25 and 10.<br />
Add <strong>the</strong> information to <strong>the</strong> graph.<br />
Point to <strong>the</strong> red dots at (11, 1), (12, 2), (12.5, 2.5), (13, 3), (14, 4), …, and (20, 10).<br />
T: Why are <strong>the</strong> dots along this diagonal red?<br />
S: They are for <strong>the</strong> cases where <strong>the</strong> middle piece of <strong>the</strong> stick would be exactly 10 cm long.<br />
G-74<br />
IG-VI
G 1 6<br />
By now <strong>the</strong> class should suspect that solid areas of red and blue, as suggested by <strong>the</strong> red-blue<br />
segments, can be colored in. Shade <strong>the</strong> appropriate regions.<br />
T: What is <strong>the</strong> probability that a fly landing in <strong>the</strong> honey triangle will land in <strong>the</strong> blue region?<br />
What is <strong>the</strong> probability that if two breaking points are chosen randomly, <strong>the</strong> pieces will<br />
form a triangle?<br />
S: 1<br />
⁄4; <strong>the</strong> area of <strong>the</strong> blue region is one fourth of <strong>the</strong> area of <strong>the</strong> honey triangle.<br />
IG-VI G-75
G-76<br />
IG-VI